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- A meshfree method with domain decomposition for Helmholtz boundary value problemsPublication . Valtchev, SvilenIn the framework of meshfree methods, we address the numerical solution of boundary value problems (BVP) for the non-homogeneous modified Helmholtz partial differential equation (PDE). In particular, the unknown solution of the BVP is calculated in two steps. First, a particular solution of the PDE is approximated by superposition of plane wave functions with different wavenumbers and directions of propagation. Then, the corresponding homogeneous BVP is solved, for the homogeneous part of the solution, using the classical method of fundamental solutions (MFS). The combination of these two meshfree techniques shows excellent numerical results for non-homogeneous BVPs posed in simple geometries and when the source term of the PDE is sufficiently regular. However, for more complex domains or when the source term is piecewise defined, the MFS fails to converge. We overcome this problem by coupling the MFS with Lions non-overlapping domain decomposition method. The proposed technique is tested for the modified Helmholtz PDE with a discontinuous source term, posed in an L-shaped domain.
- On the application of the method of fundamental solutions to boundary value problems with jump discontinuitiesPublication . Alves, Carlos J.S.; Valtchev, SvilenTwo meshfree methods are proposed for the numerical solution of boundary value problems (BVPs) for the Laplace equation, coupled with boundary conditions with jump discontinuities. In the first case, the BVP is solved in two steps, using a subtraction of singularity approach. Here, the singular subproblem is solved analytically while the classical method of fundamental solutions (MFS) is applied for the solution of the regular subproblem. In the second case, the total BVP is solved using a variant of the MFS where its approximation basis is enriched with a set of harmonic functions with singular traces on the boundary of the domain. The same singularity-capturing functions, motivated by the boundary element method (BEM), are used for the singular part of the solution in the first method and for augmenting the MFS basis in the second method. Comparative numerical results are presented for 2D problems with discontinuous Dirichlet boundary conditions. In particular, the inappropriate oscillatory behavior of the classical MFS solution, due to the Gibbs phenomenon, is shown to vanish.
- Extending the method of fundamental solutions to non-homogeneous elastic wave problemsPublication . Alves, Carlos J. S.; Martins, Nuno F. M.; Valtchev, SvilenTwo meshfree methods are developed for the numerical solution of the non-homogeneous Cauchy–Navier equations of elastodynamics in an isotropic material. The two approaches differ upon the choice of the basis functions used for the approximation of the unknown wave amplitude. In the first case, the solution is approximated in terms of a linear combination of fundamental solutions of the Navier differential operator with different source points and test frequencies. In the second method the solution is approximated by superposition of acoustic waves, i.e. fundamental solutions of the Helmholtz operator, with different source points and test frequencies. The applicability of the two methods is justified in terms of density results and a convergence result is proven. The accuracy of the methods is illustrated through 2D numerical examples. Applications to interior elastic wave scattering problems are also presented.
- Hardware-in-the-loop system numerical methods evaluation based on brush DC-motor modelPublication . Mudrov, Mikhail; Zyuzev, Anatoliy; Konstantin, Nesterov; Valtchev, Stanimir; Valtchev, SvilenHardware-in-the-loop (HiL) systems nowadays become popular. During the HiL creation process it is important to select a proper numerical method because the accuracy of the process simulation, in case of the HiL, depends on the correct numerical approach selection. That is why it is important to evaluate the most popular numerical approaches. The “Sequential” Euler (solves equations step-by-step), the Forward Euler (the most popular method for HiL creators), and the Second-order Adams-Bashforth approach are under consideration. A mathematical model based on the DC-motor with independent excitation is selected as a basis for the experimental numerical calculation. In consequence, recommendations for the numerical method selection are given, based on the obtained results.